Some definitions: A subfactor is an inclusion of factors. A factor is a von Neumann algebra with a trivial center. The center is the intersection with the commutant. A von Neumann algebra is an algebra of bounded operators on an Hilbert space, closed by taking bicommutant and dual. Here, $R$ is the hyperfinite $II_{1}$ factor. $R^{G}$ is the subfactor of $R$ containing all the elements of $R$ invariant under the natural action of the finite group $G$. In its thesis, Vaughan Jones shows that, for all finite group $G$, this action exists and is unique (up to outer conjugacy, see here p8), and the subfactor $R^{G} \subset R$ completely characterizes the group $G$. See the book Introduction to subfactors (1997) by Jones-Sunder.

What is a "subfactor"? It might help if you defined the notation $R^G$.
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Vidit NandaJul 7 '13 at 17:55

Ok @ViditNanda, I have edited some definitions and an hyperlink to The book.
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Sébastien PalcouxJul 7 '13 at 19:04

Sebastien: In order to help people who don't know what factors are understand your question, you might want to include some purely group-theoretic consequences of your notion of equivalence. Alternatively, you might want to include some examples of pairs H1⊂G1 and H2⊂G2 that are equivalent in your sense.
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André HenriquesJul 7 '13 at 19:35

3 Answers
3

For finite groups, the answer was given by Izumi in his paper "Characterization of isomorphic group-subgroup subfactors" (MR1920326). There he looks at the crossed product subfactor, but you can always take duals.

Thank you Dave for this answer about finite groups.
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Sébastien PalcouxJul 8 '13 at 20:34

1

Dave: It would be even more useful if you could state the answer here, instead of just giving a pointer to the literature.
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André HenriquesJul 8 '13 at 21:15

I don't read its proof about the classification of subfactors at index 5 in this paper. Do you know if it is published somewhere ? My original motivation is about index 6 (see here), in particular, the list of maximal subgroups at index 6. Is it known ?
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Sébastien PalcouxJul 9 '13 at 9:50

This is somewhere between a comment and an answer. But it is too long for a comment, so I put it here.

To me the natural thing to look at is the action $G\curvearrowright G/H$. $|G/H|$ captures the index, which you surely want to do, and the action should in some sense capture the position of $H$ inside $G$. The issue then would be to try to come up with an appropriate notion of equivalence of these two objects. Here are 3 possibilities:

So let $H\subset G$ and $\Lambda\subset \Gamma$ be two inclusions.

1.) We can say $H\subset G \simeq_1 \Lambda\subset\Gamma$ if there is an isomorphism of groups $\Phi:G\rightarrow\Gamma$ such that $\Phi(H)=\Lambda$. This is the strongest notion of equivalence and would certainly imply that the two actions on the cosets spaces are the "same". The issue is that it requires the groups involved to be isomorphic, which you certainly don't want.

2.) We can forget about the acting group per se and only consider the action on the coset space (in particular look at the orbit equivalence relation). We define this equivalence relation by saying $g_1H\simeq g_2H\Leftrightarrow \exists g\in G$ such that $gg_1H=g_2H$. We do a similar thing on $\Gamma/\Lambda$. Then we say $H\subset G \simeq_2 \Lambda\subset\Gamma$ if there exists a bijection $\Phi:G/H\rightarrow \Gamma/\Lambda$ that takes equivalence classes to equivalence classes.

3.) The last one is exactly what you suggest for subfactors.

Just a quick final few comments. $1\Rightarrow 2$ but I'm not sure if $2\Rightarrow 3$. The motivation for these comes from three notions of equivalence you can give to a measure preserving action of a discrete group. In this case (for the analogous three equivalences) we have $1\Rightarrow 2\Rightarrow3$ and none of the implication are reversible, in general. Much of Popa's deformation/rigidity program is in trying to reverse these arrows in certain cases. In the ergodic theory case, $1\Rightarrow 2$ is obvious and $2\not\Rightarrow 1$ is not to hard. $2\Rightarrow 3$ isn't to hard either but $3\not\Rightarrow 2$ was quite difficult.

Thank you Owen for this detailed overview. Do you have some references for these results ? The answer of Dave gives a reference with an explicit group-theoretic characterization of the third possibility for finite groups. Do you suggest (through your answer) there is no such characterization in general ?
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Sébastien PalcouxJul 8 '13 at 20:52

For the ergodic theory motivation, check out Popa's ICM talk math.ucla.edu/~popa/ICMpopafinal.pdf. There is much more there than I have suggested here. As far as what is true in general I have not given it thought. Though I should say that I left out one additional notion of equivalence. Namely isomorphism of the inclusion $L(H)\subset L(G)$ of group von Neumann algebras. Note that for countably infinite discrete groups the nicest case would be when both groups are icc, then you get $II_1$ factors. In the finite index case I don't know any references for the standard invariant.
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Owen SizemoreJul 8 '13 at 21:18

Do you know if $3\not\Rightarrow 2$ is also true for finite groups ?
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Sébastien PalcouxJul 9 '13 at 14:44

Corollary: $\sim_2$ $\Leftrightarrow$ $\sim$ if the inclusions are maximal.Problem: Extension to all the "natural" inclusions (see the definition in the optional part of this post).
The Kodiyalam-Sunder counter-examples are not "natural" inclusions, because they are single chain but not homogeneous single chain.